Let S be the set of all points in `(-pi, pi)` at which the f(x)=min(sinx ,cosx) is not differentiable Then, S is a subset of which of the following?

Options

A. `{-(3pi)/4,-pi/4,(3pi)/4,pi/4}`
B. `{-(3pi)/4,-pi/2,(pi)/2,(3pi)/4}`
C. `{-(pi)/2,-pi/4,(pi)/4,(pi)/2}`
D. `{-pi/4,0,pi/4}`

Answer

To solve the problem, we need to find the set \( S \) of all points in the interval \( (-\pi, \pi) \) where the function \( f(x) = \min(\sin x, \cos x) \) is not differentiable.

### Step-by-Step Solution:

1. **Understanding the Functions**:
- We need to analyze the functions \( \sin x \) and \( \cos x \) over the interval \( (-\pi, \pi) \).
- The function \( f(x) \) will take the value of the lower of the two functions at any given point.

2. **Finding Points of Intersection**:
- To determine where \( f(x) \) changes from \( \sin x \) to \( \cos x \) (or vice versa), we need to find the points where \( \sin x = \cos x \).
- This occurs when \( \tan x = 1 \), which gives us the solutions:
\[ x = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
- Within the interval \( (-\pi, \pi) \), the relevant points of intersection are:
\[ x = -\frac{3\pi}{4} \quad \text{and} \quad x = \frac{\pi}{4} \]

3. **Analyzing the Behavior of \( f(x) \)**:
- In the intervals defined by these points:
- For \( x < -\frac{3\pi}{4} \): \( \sin x < \cos x \) → \( f(x) = \sin x \)
- For \( -\frac{3\pi}{4} < x < \frac{\pi}{4} \): \( \cos x < \sin x \) → \( f(x) = \cos x \)
- For \( x > \frac{\pi}{4} \): \( \sin x < \cos x \) → \( f(x) = \sin x \)

4. **Identifying Non-Differentiable Points**:
- The function \( f(x) \) will not be differentiable at the points where it transitions from one function to another, which are the points of intersection found earlier:
- \( x = -\frac{3\pi}{4} \)
- \( x = \frac{\pi}{4} \)

5. **Conclusion**:
- The set \( S \) of points where \( f(x) \) is not differentiable is:
\[ S = \left\{ -\frac{3\pi}{4}, \frac{\pi}{4} \right\} \]

6. **Subset Determination**:
- Since we are asked to find out which of the given options contains the points in \( S \), we check the options provided (not specified here).
- The correct option will be the one that includes both \( -\frac{3\pi}{4} \) and \( \frac{\pi}{4} \).